FIN $666-$Advanced Quant Methods and Machine Learning in Finance

Week $3$ Homework Questions:

Part I: Problems $(20\%)$

If A and $B$ are perfectly correlated, then what is the volatility of a portfolio that is $50\%$ in $A$

and $50\%$ in $B?$

The volatility of a portfolio of two assets that have non$-$zero weights and a correlation

coefficient of $0\mathrm{.}9$ is always strictly lower than the weighted average volatility of the two

assets. True or false? Explain.

An analyst tells you that the minimum$-$volatility portfolio you can construct from two risky

assets $x$ and $Y$ is to put $100\%$ of your capital in $x.$ What does this tell you about the

correlation between the two assets?

All possible portfolios constructed from three assets $x,Y$ and $Z$ must lie on a curve between

$x$ and $Z$ that passes through $Y.$ True or false? Explain.

Write down and expand the expressions for return and variance of a portfolio of two assets.

Write down similar expressions for a portfolio of three assets.

You are a pension fund manager and you are considering two mutual funds with the

following probability distribution. Assume that the two assets have a correlation coefficient

of $0\mathrm{.}25$ and the risk free rate is $3\%.$

a$)$ What are the weights on each asset in the minimum$-$variance portfolio that consists of

these two funds?

b$)$ What is the Sharpe ratio of the minimum$-$variance portfolio?